I'm actually writing an implementation of Random Forests but I believe the question is specific to decision trees (independent of RFs).
So the context is that I'm creating a node in a decision tree and both the prediction and target variables are continuous. The node has a split threshold to partition data into two sets, and I create a new prediction for each subset based on the average target value in each set. Is this the correct approach?
The reason I ask is that when predicting binary variables I believe the typical (correct?) approach is to divide the data into 0 and 1 subsets without taking an average over the data rows in each subset. Subsequent splits will divide into finer grained subsets and taking an average at each split results subsequent splits (lower down the decision tree) operating on what are now continuous variables rather than binary variables (because we are operating on the residual error values instead of the original targets).
Side question: Is the distinction between the two approaches (binary vs continuous) significant - or will they actually give identical results for a complete decision tree?